APPLICATION OF COMBINATORIAL METHODS TO PROTEIN
IDENTIFICATION IN PEPTIDE MASS FINGERPRINTING
Leonid Molokov
Department of Computer Science and Engineering, Chalmers University of Technology, Gothenburg, Sweden
Keywords:
Bioinformatics, Proteomics, Peptide mass fingerprinting, Error removal, Union editing, Set cover, Hitting set,
Simulations.
Abstract:
Peptide Mass Fingerprinting (PMF) for long has been a widely used and reliable method for protein identifica-
tion. However it faced several problems, the most important of which is inability of classical methods to deal
with protein mixtures. To cope with this problem, more costly experimental techniques are employed. We
investigate, whether it is possible to extract more information from PMF by more thorough data analysis. To
do this, we propose a novel method to remove noise from the data and show how the results can be interpreted
in a different way. We also provide simulation results suggesting our method can be used for analysis of small
mixtures.
1 INTRODUCTION
Proteomics is an important and fast developing field
in bioinformatics, concerned in protein identifica-
tion and quantification. As proteins are complex
molecules, often one can only measure the proper-
ties of their side-products or components. Proteomics
studies how to produce smaller molecules, what prop-
erties of theirs to investigate and how to recover infor-
mation about initial proteins from these known prop-
erties. One of the most extensively used techniques
in the field is peptide mass fingerprinting (PMF). The
key idea of the method is to break proteins into pieces,
measure masses of the fragments and use them to
identify the initial proteins. Protein fragmentation is
believed to preserve through repeating experiments
thus making possible analysis of the observed set of
masses. Although in past decades the method was
widely used, now it loses to shotgun proteomics, be-
cause of these two major problems:
1. Mass ambiguity. There are peptides with nearly
identical masses, with difference being less than a
sensitivity of the measurement tool.
2. Single protein analysis. Many widely-used anal-
ysis methods such as Mascot (Perkins et al.,
1999), Ms-Fit (Clauser et al., 1999) and ProFound
(Zhang and Chait, 2000) treat the sample as if it
was a single protein. Analysis of more complex
mixtures is deemed to be too difficult due to rea-
sons that we are going to discuss below.
Mass ambiguity leads to less specificity in protein
identification, and is an experimental setup problem,
which we hardly can solve. On the contrary, single
protein limitation is put by the method interpreting
the data, what was already recognized in earlier works
(He et al., 2009),(Jensen et al., 1997) with attempts to
provide solutions for protein mixtures.
But why was this limitation put? Consider a set
of masses M = {m
1
, m
2
, . . . , m
k
} observed in the ex-
periment. Let us make an assumption that the exper-
iment was held in perfect conditions and each partic-
ular mass m corresponds to some protein p, whose
mass spectrum M
p
⊂ M, is fully represented in the
experimental spectrum. It is also reasonable to put
some limitations on the cardinality of initial set of
proteins P = {p
1
, p
2
, . . . , p
n
}, that produced spectrum
M: we assume n ≤ c (where c is some integer con-
stant). We also have a collection of theoretical spec-
tra M
p
, p ∈ P
db
for each protein p in database. One
can see that the question, whether there exist a set of
proteins P
s
such that card(P) ≤ c and M ⊆
S
p∈P
s
M
p
explaining the spectrum M, is precisely a SET COVER
problem, which is known to be NP-complete. We
have an enumeration problem, i. e. describe all the
possible satisfying sets (if there are any) which is
even harder. Nevertheless, simulations show that in
practical cases, most proteins have unique masses,
what means quite optimistic real running time. We
will cover that in section ”Simulation”. Worst case
analysis is not hopeless as well, if one fixes the
307
Molokov L..
APPLICATION OF COMBINATORIAL METHODS TO PROTEIN IDENTIFICATION IN PEPTIDE MASS FINGERPRINTING.
DOI: 10.5220/0003102703070313
In Proceedings of the International Conference on Knowledge Discovery and Information Retrieval (KDIR-2010), pages 307-313
ISBN: 978-989-8425-28-7
Copyright
c
 2010 SCITEPRESS (Science and Technology Publications, Lda.)

maximum number of occurrence (degree) of an el-
ement (mass) in different sets (proteins), the prob-
lem is fixed-parameter tractable (FPT), for details,
see (Damaschke and Molokov, 2009). It seems that
masses with high degrees are not informative. In
extreme case where a degree of particular mass is
equal to number of all possible proteins (omnipresent
mass), it can be safely removed from problem formu-
lation, as any protein mixture will explain it. Other-
wise, almost for sure there exists a mass that explains
a subset of proteins that mass with big degree does.
What leads us to conclusion, that one can drop high-
degree masses from consideration (and thus bound the
the maximum degree of an element). Also, obtained
solutions can be verified later to explain these masses.
The aforementioned computational problem sur-
prisingly reappears in works dedicated to protein in-
ference in shotgun proteomics, (Nesvizhskii and Ae-
bersold, 2005; Aebersold and Mann, 2003), where in-
stead of mass spectra, the peptide sequences are used.
Again the goal is identification of proteins, and the
problem was recognized as NP-hard in (Aebersold
and Mann, 2003).
One can see, that in both cases, whether the
proteins are being identified by observed masses or
peptides sequences, the presence of errors should
be taken into account. Indeed, some observations
(masses, sequences) could be lost or altered the way
that equipment cannot detect them, and some repre-
sent noisy peaks, peptide fragments that were inserted
as a consequence of flaws in the experiment. A sig-
nificant amount of noise can be removed as described
in (Samuelsson et al., 2004), but these procedures do
not guarantee error-free output. A way to fix it is to
reformulate the problem as SET COVER WITH MISS-
ING ELEMENTS : given a spectrum M, a database
M
p
, p ∈ P
db
of theoretical mass spectra and numbers
k, c, find out if there’s a set of proteins P
s
s. t.
card(P) ≤ c, |M△
[
p∈P
s
M
p
| ≤ k.
Let us refer to it in future as SCME. If k = 0, this is a
classical SET COVER problem.
Unfortunately if k ≥ 10, resulting space of pos-
sible solutions contains so many sets and elements,
that it is infeasible to find solutions in reasonable
time. It might be the reason why scoring candidate
set is preferred to exact enumeration of solutions in
both classical single-protein schemes and in attempts
to cope with more complex mixtures, (He et al.,
2009),(Jensen et al., 1997). There is an implicit as-
sumption that there exists a set of proteins (or a single
protein) that explains the data best; hence, a scoring
function is introduced, and a set of proteins on which
the function reaches its optimal value is searched (un-
fortunately, it has an expected problem, that only local
optimum can be guaranteed). We considered a differ-
ent possibility, that there might be several such sets,
and it might be interesting to observe all possible so-
lutions, to see, e.g., which proteins present in all or in
most of solutions (omnipresent proteins), which seem
to appear one instead of another, and which are only
observed in a few solutions. The first type of proteins
are the most important ones, because if one assumes
that initial set of proteins is among listed solutions,
then omnipresent proteins are contained in initial set
of proteins as well.
One way to reduce the complexity of SCME is to
use some extra information from the data. We already
know that without errors, the mass spectrum should
be precisely the union of the theoretical spectra of
some set of proteins. This knowledge gives rise to
a different approach: first, remove the errors, then try
to solve the now simple (according to what simula-
tions show) problem. To remove errors, we want to
solve another computational problem, namely UNION
EDITING, which we formulate below: Given set M, a
collection of its subsets M
p
, p ∈ P
db
and integers k
and l, find some complete union M
′
which is obtained
from M by at most k insertions and l deletions, that is,
|M
′
\ M| ≤ k, |M \ M
′
| ≤ l.
(A complete union notion will be defined in ”Formal-
ization” section for the sake of consistency.) Again, in
the enumeration version we want all such sets M
′
. It
appears that a simple branching algorithm for solv-
ing UNION EDITING, that we will show later, works
successfully and sufficiently fast.
Therefore, the approach of enumerating the pos-
sible satisfying protein sets seems promising to us.
We are going to present algorithm solving the prob-
lem with simulation results saved for the last section
of this article.
2 FORMALIZATION
2.1 Preprocessing
It is time go into more details about how one would
formulate the biological problem in mathematical
terms. We will refer to protein set (mixture) P =
{p
1
, p
2
, . . . , p
k
} as initial protein set. As a result of
mass spectroscopy, P generated a set of masses M =
{m
1
, m
2
, . . . , m
c
} (theoretical spectrum of P), which
was changed by experimental errors into M
′
(empir-
ical spectrum of P). We want to understand what P
was.
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308

Let us say that P (set of proteins) explains M (the-
oretical spectrum of P), if M =
S
p∈P
M
p
. If, however,
there is a P
′
⊂ P, that explains M fully, i. e. produces
the same spectrum, it is impossible to say, given the
data, whether P or P
′
generated M. We will use Oc-
cam’s Razor principle in a similar way as in (Nesvizh-
skii and Aebersold, 2005). More precisely, we think
it is reasonable to seek only minimal protein sets, that
is, sets with the property that if one vertex is removed
from the protein set, it no longer explains M. It is also
clear, that one has to limit the size of candidate mix-
tures, because experimental setup attempts to reduce
the number of proteins under investigation.
As was mentioned before, in our model we as-
sume that throughout the experiment, ideal spectrum
M was altered by k insertions and and l deletions. Let
us denote set of inserted masses by M
ins
and set of
deleted masses by M
del
(M
d
el ⊆ M). What we in real-
ity observe is empirical spectrum M
′
, from which we
want to infer what possible spectra M could be. Some
insertions can be removed almost immediately, for
the first preprocessing steps, see (Samuelsson et al.,
2004), however, some will still remain. Unless we
specify boundaries for possible amounts of errors (re-
flecting the quality of experiment), it is impossible to
say, what M was, because M
′
could drift too far from
it’s ancestor. I.e. we have to make the additional
assumption, that k and l are limited by constants k
0
and l
0
. As soon as we do it, however, we get an im-
mediate outcome: none of the proteins P
big
: {p ∈
P
db
|card(M
p
\M
′
) > l
0
or card(M
′
\M
p
) > k
0
} were
in P. It is a useful observation, because if a mass can
be only produced by proteins in P
big
, it could not be
in M.
In other words, we have a hypergraph H, whose
vertices represent masses, and each of its edges rep-
resents protein. A vertex representing a mass is inci-
dent to an edge representing a protein iff the protein
has the mass in its spectrum. Essentially, H = (V, E),
where V = M
all
, E = P
db
, where M
all
represent all
the masses in the database. Given M
′
⊆ M
all
, a set of
observed masses, k
0
and l
0
as integer boundaries for
the numbers of insertions and deletions respectively,
find M ⊆ M
all
such that |M\M
′
| < l
0
, |M
′
\M
p
< k
0
|
and M =
S
p∈P
s
M
p
, where P
s
is some subset of P
db
.
In this error correction problem we do not limit the
size of P
s
by constant c, as it can be done on a later
stage, where one solves the Set Cover problem for the
recovered masses spectra.
As was said, the list of possible M for H and for
H
′
= (M
all
, P
db
\P
big
) is the same (follows immedi-
ately from problem formulation), what gives substan-
tial reduction of edges. Can we do anything else? The
answer is yes, but we have to make an additional as-
sumption. Consider a mass (vertex) m ∈ M
all
. If m
is not adjacent to any mass (vertex) from M in hyper-
graph H
′
, then it is one of thousands of other masses
in the database, about presence of which we have no
available evidence. We assume that no such mass was
initially in the experiment, because we cannot make
any definite conclusion about these masses. It will al-
low us to shrink the hypergraphto edges adjacent only
to M
′
. Let us denote such a hypergraph H
′
(M
′
).
The aforementioned assumption is strong and can
lead to exclusion of some proper solutions. How dras-
tic is the difference to full list of solutions? Imagine
we indeed have M
iso
⊆ M : M
iso
∩
S
e∈H
′
(M
′
)
e =
/
0.
I.e. edges of H(M
′
) and H(M
iso
) do not intersect
(they form different connectivity components). Let
us imagine we are solving the full problem, consider
hypergraph H(M
′
) ∪ H(M
iso
). After union editing,
which can only add or remove vertices, these two
components will be still disconnected, what means
that the SET COVER instance will contain also two
connectivity components in its hypergraph. It is ob-
vious that every solution for this instance of SET
COVER can be split into two parts, each of which ex-
plaining one component. If a solution part size for
H(M
iso
is small and fixed, solutions for another com-
ponent will represent a part of some solution for the
whole hypergraph. It suggests that as a result of our
limitation, we can only lose some proteins from the
solutions, but we can’t get any extra unwanted pro-
teins.
To summarize, we have made the following as-
sumptions:
• The mixture size is limited by constant, i. e. Set
Cover size is limited by parameter c.
• The true mixture was altered by a limited number
of k
0
insertions and l
0
deletions
• Proteins are observed at least partially, i. e. there
are no entirely lost proteins (if so, we simply ig-
nore them). This does not affect what we learn
about other proteins.
2.2 Algorithm for Union Editing
Enumeration
In this subsection we will give a simple branching
algorithm that is able to solve the UNION EDITING
problem.
Let us call a vertex v of some set M
′
complete in
hypergraph H = (V, E) (where not necessarily M
′
⊆
V), if ∃e ∈ E s.t. v ∈ e, e ⊆ M
′
. Remember, that we
are interested in sets M
′
that are a union of some
subfamily of M
p
, p ∈ P
db
, let us call such sets com-
plete unions. It follows from definition that M
′
is
APPLICATION OF COMBINATORIAL METHODS TO PROTEIN IDENTIFICATION IN PEPTIDE MASS
FINGERPRINTING
309

complete union iff every vertex v ∈ M
′
is complete.
That immediately gives an idea of a branching algo-
rithm: to guarantee that a particular internal vertex
v is complete, we have to either remove v from M
′
;
or choose some edge e, v ∈ e and add all its external
vertices (that were not in M
′
). Whenever we remove
the vertex, we assume it was added by mistake, thus
we should count this occurrence as a separate inser-
tion. Same way, when we add the vertex, we consider
it was deleted erroneously, and thus we should count
it as a deletion. Therefore, throughout the branching
process, we shall maintain counters for allowed inser-
tions and deletions (k and l are given in the problem
instance). The moment some counter turns to be neg-
ative, the current branch should be terminated.
Pseudocode of the algorithm is given below:
Input: hypergraph H = (V, E) ,
integers k , l (insertions, deletions),
set of initial vertices M ⊆ V .
Find all complete unions M
′
:
M
′
⊆ V , |M \ M
′
| ≤ k and |M
′
\ M| ≤ l .
begin
FindCompleteUnion( H , k , l , M , M
′
)
end
FindCompleteUnion( H , k , l , M , M
′
)
begin
if M =
/
0 and 0 ≤ kl
output M
′
terminate
end if
while k ≥ 0 and l ≥ 0 do
choose arbitrary vertex v in graph H
B :=
/
0
for all edges e ∈ E incident to v
A
′
:= { vertices incident to e} \ M
FindCompleteUnion( H
′
(V, E) , k ,
l − |A
′
| , M , M
′
∪ A
′
)
B := B∪ {e}
end for
FindCompleteUnion( H
′
(V, E\B) ,
k− 1 , l , M\{v} , M
′
\{v} )
end while
end
Note that we stop whenever we find any feasible solu-
tion, even if k and l are not exhausted. By doing that,
we stick to the previously chosen approach: find only
minimal solutions of the instance. But what would
minimality mean in this case?
In the introduction section, we mentioned, that we
want to use only information coming from the empir-
ical spectrum. What means, if ∃e ∈ E : e ∩ {v} =
/
0,
e ⊆ M
′
, i.e. if there is an edge in M’ that is not ex-
plained by an initial set of vertices M, then M
′
is not
a minimal solution. Note, that it can be obtained from
solution that does not have such edges by several re-
movals of vertices - one can say, that non-minimal
solution assumes more deletions then necessary.
Imagine an opposite situation, that a solution adds
more insertions, than it should be. Keeping in mind
that all solutions are complete unions, we can con-
clude that all vertices which are removed in non-
minimal solution, were complete (we remove only
vertices from M, and if vertex v ∈ M was not com-
plete, then it should be either completed by other ver-
tices or removed in every possible solution). Hence,
we obtain another condition: none of the initially
complete vertices should be removed.
Definition 1. Solution M
′
of instance of UNION
EDITING (H = (V, E),k, l, M) is not minimal if ∃M
′′
— another solution, such that (M
′′
= M
′
\e, e ∈ E,
e∩M =
/
0) or (M
′′
= M
′
∪ e, e ∈ E, e ⊆ M). All other
solutions are minimal.
Theorem 1. FindCompleteUnion(H, k, l, M, M) out-
puts solutions of (H, k, l, M) instance of UNION
EDITING and among them lists all minimal solutions.
We do not mention any upper bounds on running
time. It can be shown that CLIQUE can be reduced
to a special case of UNION EDITING, thus the prob-
lem is (not surprisingly) NP-hard. What means it will
be hard to get optimistic bounds for the general case.
For our inputs though the algorithm runs fast enough
to forget about computational complexity (which it-
self is an interesting separate topic to discuss for this
problem), and these results we are going to cover in
the next section.
3 SIMULATIONS
The main question of the simulations was whether it
is feasible to get a list of minimal solutions for regu-
lar problem instance in reasonable time. We are also
interested in the false positives and in the recall (here,
the rate of detected proteins from the initial protein
set), as the practical motivation dictates us to maxi-
mize the latter and get rid of the former.
To answer these questions, we generated uni-
formly at random 10 initial protein sets of sizes 5,
10, 15 and 20. Then we obtained a theoretical mass
spectrum from our database (see ”Data” subsection)
which we modified by deleting 1 ...9 of the theoret-
ical masses uniformly at random and then inserting 1
...9 masses, also chosen uniformly at random. Here
our simulation approach significantly deviated from
standard (well described in (He et al., 2009)), where
random numbers are inserted instead of masses from
the database and masses are modified by noise instead
of being deleted. The noise rate is, of course, higher,
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310

reaching 50 % of noise peaks. One can translate this
procedure into ours by accepting masses within some
sliding window: i.e. consider mass m observed if
∃m
′
∈ M
all
: |v(m)−v(m
′
)| < d, where v(m) represent
the actual mass value and d is the sliding window pa-
rameter, M
all
is the set of all known masses. This way
we will get also the initial spectrum modified by few
insertions and deletions but the distribution of these
insertions/deletions will be different.
We did not exclude the masses outside of de-
tectable range (usually around 500-5000 Da). Of
course, it gave us more specificity, because we should
have also removed them from the database, thus mak-
ing protein theoretical spectra less distinguishable.
However, it is rather an equipment problem, than
computational, because if we lose elements and thus
decrease the cardinality of sets in an instance of ei-
ther SET COVER or UNION EDITING, they cannot be-
come harder. And we still would list all the possible
solutions. So the undetectable masses were kept to
increase the computational challenge of the problem.
As soon as the empirical sample is generated, we
apply an implementation of algorithm described in
previous section to it. Because error bounds are in
real cases unknown, we used maximal parameter and
twice the maximal parameter as a bound. I.e. if 3
insertions and 0 deletions were introduced, we gave 6
insertions and 6 deletions as input. We also paid at-
tention to the few insertions (0 ...3) case as it seem to
describe best the real situation - noise can be filtered
in many ways, including knowledgeof the protein ori-
gin. E. g. if we know that the mixture contained only
yeast protein, then masses corresponding to peptides
from other organisms can be removed, etc.
Although the initial plan was to formulate outputs
of UNION EDITING instances as instances of SET
COVER problem and solve them by branching algo-
rithm described in (Damaschke and Molokov, 2009),
it turned out later that this is not the best idea. As the
number of the obtained solutions (possible theoretical
spectra) for UNION EDITING instances with high er-
ror bounds is too big, we would waste too much time
on going through all of them. First we need to do
some categorization of errors, as there are might be
omnipresent masses, etc. - the same way it was with
proteins. It is still an open question, how one can ex-
actly do it, but it seems rather time consuming to find
set covers for all possible theoretical spectra. Accord-
ing to running times, given in tables 1 and 2, exclud-
ing the time spent for sample preprocessing, each full
run would take more than a day!
Instead, we solve the SET COVER only for the ini-
tial protein sets. More precisely, as an input we used:
universe M (union of mass spectra of initial proteins),
and sets {p ∈ P
db
|M
p
∩ M} as covering sets. Param-
eter c is not known and should be selected by the in-
vestigator. We used value 20 for it.
We will cover the results of both algorithms in
subsection ”Performance”.
3.1 Performance
For all drawn samples, SET COVER instances ap-
peared to be trivial - they were solved in a matter
of seconds. The reason behind it was that proteins
had few shared masses and there were many unique
masses - the ones that appear only in one protein
among chosen candidates. Results for one of the sam-
ples are given in table 1.
The running times of implementation of algorithm
from (Damaschke and Molokov, 2009), suggest that
we can use it as a tool to solve SET COVER instances
on a regular basis.
Table 1: Results of implementation of the branching algo-
rithm for SET COVER enumeration.
|P| c Time Recall
20 20 1.934s 85%
15 20 2.075s 80%
10 20 0.982s 80%
5 20 0.244s 80%
In this table, |P| represents cardinality of the initial
set of proteins (sample size), c represents a parameter,
the bound for solution size, Time stands for running
time and Recall is obtained as a rate of detected pro-
teins in initial set of proteins. But what proteins are
counted as detected? If we consider only ones that are
present in all solutions as hits, then as was said in in-
troduction, we can guarantee that every such protein
was in initial set of proteins. Therefore, the number of
false positives obtained by our method will be 0 (it is
not provided in table). Recall, however, reduces. Al-
though we list the initial set of proteins in the solution,
not all it’s proteins are omnipresent.
The following table summarizes results of the
branching algorithm given in previous section.
Here ”Sample” column contains sample ids, k
and l are numbers of actual insertions and deletions,
whilst k
0
and l
0
are bounds specified for them.
First part of the table demonstrates the depen-
dency between running time and the number of initial
proteins |P|. Although there were no insertions made
at all, high bound for their number lead to high run-
ning time, which rapidly increases with growing sam-
ple size. Second part of the table proves that it is not
the actual number of errors that is important for the
running time, but the error bounds we provide. E. g.
APPLICATION OF COMBINATORIAL METHODS TO PROTEIN IDENTIFICATION IN PEPTIDE MASS
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311

Table 2: Results of implementation of the branching algo-
rithm for UNION EDITING.
Sample k l |P| k
0
l
0
Time
1 0 9 5 9 9 0.172s
1 0 9 10 9 9 19s
1 0 9 15 9 9 1102s
1 0 9 20 9 9 3572s
1 0 4 15 8 8 1260s
3 6 6 15 6 6 181s
2 9 9 5 9 9 0.634s
2 9 9 5 18 18 39s
for 12 errors in sample 3, we assume only 12 (6 inser-
tions and 6 deletions), compared to 18 (9 insertions
and 9 deletions) in previous part of the table. Third
part of the table gives the most optimistic results. If
|P| is sufficiently small, even with high error bounds,
reaching 36 - approximately 20% of initial sample
size, running time is within one minute range. That
means that our approach can be used directly for clas-
sical PMF, where a low number of proteins in mixture
is guaranteed by separation procedures.
3.2 Discussion
Results raise several questions, which are not yet an-
swered. The first one and the most important would
be: is it possible to reduce the running time for big
initial protein sets? It is clear we will get some ex-
ponential function, but the base of exponent could be
improvedby intelligent branching rules. We may con-
sider using techniques similar to ones used for solving
HITTING SET problem in (Fernau, 2006). We may
also save time by finding a way to browse through so-
lution space (list of possible solutions for high error
bounds reaches order of 10
4
) of UNION EDITING.
Another good question is, how well the method
behaves on real data. It is hard to assess what are
the real numbers of insertions and deletions in regular
experiments, butone could try solving the instance for
known protein mixtures.
Finally, it might be good to check whether it is
possible to reformulate problem into counting and
thus try some approximation scheme for SCME. Al-
though we can’t get a good polynomial time ap-
proximation scheme (Lund and Yannakakis, 1994),
some randomized approximation scheme might work
in polynomial time.
3.3 Data
Protein database uses protein sequences from Swis-
sProt version of 2005, with humans as species of
origin. Theoretical spectra were obtained by sim-
ulating trypsin digestion of proteins with no mis-
cleavages (what assumes that trypsin always cuts
where it should and does not miss any markers).
For computations we used regular machine with In-
tel Core2 Duo T5800 processor, 2.00GHz with 3GB
RAM available.
4 CONCLUSIONS
Simulations show that the method can be applied to
standard PMF setup with small protein mixtures to
improve the results obtained by classical methods.
There is also a perspective of application in shotgun
proteomics, which considers bigger protein mixtures.
It requires modifications of the algorithm for solving
UNION EDITING that will reduce its running time. A
more intelligent way of enumeration of outputs of the
algorithm might also provide an insight to a faster so-
lution.
ACKNOWLEDGEMENTS
Work was partially supported by Swedish Research
Council (Vetenskapsradet), grant no. 2007-6437,
”Combinatorial inference algorithms : parameteriza-
tion and clustering”. Author also would like to ex-
press gratitude to Swedish Institute for former support
and funding, and to his supervisor, Peter Damaschke,
for valuable input in research and studies.
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